What is the derivative of the area of a circle with respect to its radius?

The derivative of the area A of a circle with respect to its radius r is: $$\frac{dA}{dr} = 2\text{π}r$$ .

If the radius of a circular pool is increasing at a rate of 3 cm/min, how do you express this mathematically?

This can be expressed as: $$\frac{dr}{dt} = 3 \text{ cm/min}$$ .

How do you calculate the rate at which the volume of a sphere is increasing when the radius is 2 in?

Substituting r = 2 in into the formula $$\frac{dV}{dt} = 4\text{π}r^2 \times \frac{dr}{dt}$$ gives $$\frac{dV}{dt} = 4\text{π}(2^2)(2) = 32\text{π} \text{ in}^3/\text{sec}$$ .

How do you calculate the rate at which the area of a circular pool is increasing when the radius is 12 cm?

Substituting r = 12 cm into the formula $$\frac{dA}{dt} = 2\text{π}r \times \frac{dr}{dt}$$ gives $$\frac{dA}{dt} = 2\text{π}(12)(3) = 72\text{π} \text{ cm}^2/\text{min}$$ .

What is the relationship between the rates of change of volume and radius for a sphere?

The relationship is given by the formula: $$\frac{dV}{dt} = 4\text{π}r^2 \times \frac{dr}{dt}$$ , showing how the volume changes as the radius changes.

What is the relationship between the rates of change of area and radius for a circle?

The relationship is given by the formula: $$\frac{dA}{dt} = 2\text{π}r \times \frac{dr}{dt}$$ , showing how the area changes as the radius changes.

What is the significance of the chain rule in related rates problems?

The chain rule is significant in related rates problems as it allows us to relate the rates of change of different quantities that are dependent on each other.

How can you apply related rates to real-world problems?

Related rates can be applied to various real-world problems, such as determining how fast a shadow lengthens as an object moves, or how quickly a tank fills with water.

12/11 Related Rates Flashcards

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15 questions

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1.

FLASHCARD QUESTION

Front

What is a related rate problem?

Back

A related rate problem involves finding the rate at which one quantity changes with respect to another quantity that is also changing. These problems often involve the use of derivatives and the chain rule.

2.

FLASHCARD QUESTION

Front

What is the formula for the volume of a sphere?

Back

3.

FLASHCARD QUESTION

Front

How do you find the rate of change of volume with respect to time for a sphere?

Back

4.

FLASHCARD QUESTION

Front

What is the derivative of the volume of a sphere with respect to its radius?

Back

5.

FLASHCARD QUESTION

Front

If the radius of a sphere is increasing at a rate of 2 in/sec, how do you express this mathematically?

Back

6.

FLASHCARD QUESTION

Front

What is the area of a circle in terms of its radius?

Back

7.

FLASHCARD QUESTION

Front

How do you find the rate of change of area with respect to time for a circle?

Back

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